Binary Trees, Binary Search Trees COMP171 Fall 2006

Binary Search Trees / Slide 2 Trees * Linear access time of linked lists is prohibitive n Does there exist any simple data structure for which the running time of most operations (search, insert, delete) is O(log N)? * Trees n Basic concepts n Tree traversal n Binary tree n Binary search tree and its operations

Binary Search Trees / Slide 3 Trees * A tree is a collection of nodes n The collection can be empty n (recursive definition) If not empty, a tree consists of a distinguished node r (the root), and zero or more nonempty subtrees T 1, T 2,...., T k, each of whose roots are connected by a directed edge from r

Binary Search Trees / Slide 4 Some Terminologies * Child and Parent n Every node except the root has one parent n A node can have an zero or more children * Leaves n Leaves are nodes with no children * Sibling n nodes with same parent

Binary Search Trees / Slide 5 More Terminologies * Path n A sequence of edges * Length of a path n number of edges on the path * Depth of a node n length of the unique path from the root to that node * Height of a node n length of the longest path from that node to a leaf n all leaves are at height 0 * The height of a tree = the height of the root = the depth of the deepest leaf * Ancestor and descendant n If there is a path from n1 to n2 n n1 is an ancestor of n2, n2 is a descendant of n1 n Proper ancestor and proper descendant

Binary Search Trees / Slide 6 Example: UNIX Directory

Binary Search Trees / Slide 7 Example: Expression Trees * Leaves are operands (constants or variables) * The internal nodes contain operators * Will not be a binary tree if some operators are not binary

Binary Search Trees / Slide 8 Tree Traversal * Used to print out the data in a tree in a certain order * Pre-order traversal n Print the data at the root n Recursively print out all data in the left subtree n Recursively print out all data in the right subtree

Binary Search Trees / Slide 9 Preorder, Postorder and Inorder * Preorder traversal n node, left, right n prefix expression  ++a*bc*+*defg

Binary Search Trees / Slide 10 Preorder, Postorder and Inorder * Postorder traversal n left, right, node n postfix expression  abc*+de*f+g*+ * Inorder traversal n left, node, right n infix expression  a+b*c+d*e+f*g

Binary Search Trees / Slide 11 Example: Unix Directory Traversal PreOrder PostOrder

Binary Search Trees / Slide 12 Preorder, Postorder and Inorder Pseudo Code

Binary Search Trees / Slide 13 Binary Trees * A tree in which no node can have more than two children * The depth of an “average” binary tree is considerably smaller than N, even though in the worst case, the depth can be as large as N – 1. Generic binary tree Worst-case binary tree

Binary Search Trees / Slide 14 Node Struct of Binary Tree * Possible operations on the Binary Tree ADT n Parent, left_child, right_child, sibling, root, etc * Implementation n Because a binary tree has at most two children, we can keep direct pointers to them

Binary Search Trees / Slide 15 Convert a Generic Tree to a Binary Tree

Binary Search Trees / Slide 16 Binary Search Trees (BST) * A data structure for efficient searching, inser- tion and deletion * Binary search tree property n For every node X n All the keys in its left subtree are smaller than the key value in X n All the keys in its right subtree are larger than the key value in X

Binary Search Trees / Slide 17 Binary Search Trees A binary search tree Not a binary search tree

Binary Search Trees / Slide 18 Binary Search Trees * Average depth of a node is O(log N) * Maximum depth of a node is O(N) The same set of keys may have different BSTs

Binary Search Trees / Slide 19 Searching BST * If we are searching for 15, then we are done. * If we are searching for a key < 15, then we should search in the left subtree. * If we are searching for a key > 15, then we should search in the right subtree.

Binary Search Trees / Slide 20

Binary Search Trees / Slide 21 Searching (Find) * Find X: return a pointer to the node that has key X, or NULL if there is no such node * Time complexity: O(height of the tree) BinaryNode * BinarySearchTree::Find(const float &x, BinaryNode *t) const { if (t == NULL) return NULL; else if (x element) return Find(x, t->left); else if (t->element < x) return Find(x, t->right); else return t; // match }

Binary Search Trees / Slide 22 Inorder Traversal of BST * Inorder traversal of BST prints out all the keys in sorted order Inorder: 2, 3, 4, 6, 7, 9, 13, 15, 17, 18, 20

Binary Search Trees / Slide 23 findMin/ findMax * Goal: return the node containing the smallest (largest) key in the tree * Algorithm: Start at the root and go left (right) as long as there is a left (right) child. The stopping point is the smallest (largest) element * Time complexity = O(height of the tree) BinaryNode * BinarySearchTree::FindMin(BinaryNode *t) const { if (t == NULL) return NULL; if (t->left == NULL) return t; return FindMin(t->left); }

Binary Search Trees / Slide 24 Insertion * Proceed down the tree as you would with a find * If X is found, do nothing (or update something) * Otherwise, insert X at the last spot on the path traversed * Time complexity = O(height of the tree)

Binary Search Trees / Slide 25 Deletion * When we delete a node, we need to consider how we take care of the children of the deleted node. n This has to be done such that the property of the search tree is maintained.

Binary Search Trees / Slide 26 Deletion under Different Cases * Case 1: the node is a leaf n Delete it immediately * Case 2: the node has one child n Adjust a pointer from the parent to bypass that node

Binary Search Trees / Slide 27 Deletion Case 3 * Case 3: the node has 2 children n Replace the key of that node with the minimum element at the right subtree n Delete that minimum element  Has either no child or only right child because if it has a left child, that left child would be smaller and would have been chosen. So invoke case 1 or 2. * Time complexity = O(height of the tree)